Quantum Field Theory on Curved Backgrounds — A Primer

Abstract

Goal of this review is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give to the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.Keywords: quantum field theory on curved backgrounds, algebraic quantum field theory PACS numbers: 04.62.+v, 11.10.Cd or lightlike vector depending whether its length g(v p , v p ) is smaller, greater or equal to 0 respectively. In this way, as in Minkowski spacetime, we divide T p M into two regions, the set of spacelike vectors, and the two-folded light cone stemming from 0, the origin of the vector space T p M . For each point p ∈ M we have the freedom to designate each of the folds of the light cone of T p M as the set of future-directed and of pastdirected vectors respectively. If we can smoothly specify at each point which one of the two cones is the future one, we say that (M, g) is time orientable. 123 This is equivalent to the existence of a global vector field on M which is timelike everywhere. Henceforth we will only consider pairs (M, g) enjoying this property and, moreover, we shall assume that a time orientation has been fixed. This allows to introduce J ± M even when (M, g) is not isometric to (R 4 , η). To be precise, in the first place one has to define timelike, lightlike and spacelike curves: A piecewise smooth curve γ : I → M , I ∈ [0, 1], is timelike (lightlike or spacelike) if, for every t ∈ I, the vector tangent to the curve at γ(t) is timelike (respectively lightlike or spacelike). A curve is called causal if it is nowhere spacelike. For causal curves, one can also specify the direction according to the time orientation of (M, g): A causal curve γ : I → M is future-(past-) directed if for each t ∈ I each vector tangent to γ at γ(t) lies in the future (respectively past) fold of T γ(t) M . Given these preliminaries, for any p ∈ M we call causal future of p the set J + M (p) of points q ∈ M which can be reached by a future-directed causal curve stemming from p. Replacing future-directed causal curves with past-directed ones, we define also the causal past of p, J − M (p). Notice that, if we allow only timelike (in place of causal) curves, we replace J ± M (p) with I ± M (p), namely the chronological future (+) an...

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